3.532 \(\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=146 \[ \frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^5(c+d x)}{5 d}-\frac{a^4 \csc ^4(c+d x)}{d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}+\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{10 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d} \]

[Out]

(10*a^4*Csc[c + d*x])/d + (2*a^4*Csc[c + d*x]^2)/d - (4*a^4*Csc[c + d*x]^3)/(3*d) - (a^4*Csc[c + d*x]^4)/d - (
a^4*Csc[c + d*x]^5)/(5*d) - (4*a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d +
(a^4*Sin[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.116307, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^5(c+d x)}{5 d}-\frac{a^4 \csc ^4(c+d x)}{d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}+\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{10 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^4,x]

[Out]

(10*a^4*Csc[c + d*x])/d + (2*a^4*Csc[c + d*x]^2)/d - (4*a^4*Csc[c + d*x]^3)/(3*d) - (a^4*Csc[c + d*x]^4)/d - (
a^4*Csc[c + d*x]^5)/(5*d) - (4*a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d +
(a^4*Sin[c + d*x]^3)/(3*d)

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^6 (a-x)^2 (a+x)^6}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^6}{x^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (4 a^2+\frac{a^8}{x^6}+\frac{4 a^7}{x^5}+\frac{4 a^6}{x^4}-\frac{4 a^5}{x^3}-\frac{10 a^4}{x^2}-\frac{4 a^3}{x}+4 a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{10 a^4 \csc (c+d x)}{d}+\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}-\frac{a^4 \csc ^4(c+d x)}{d}-\frac{a^4 \csc ^5(c+d x)}{5 d}-\frac{4 a^4 \log (\sin (c+d x))}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{a^4 \sin ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.174539, size = 96, normalized size = 0.66 \[ \frac{a^4 \left (5 \sin ^3(c+d x)+30 \sin ^2(c+d x)+60 \sin (c+d x)-3 \csc ^5(c+d x)-15 \csc ^4(c+d x)-20 \csc ^3(c+d x)+30 \csc ^2(c+d x)+150 \csc (c+d x)-60 \log (\sin (c+d x))\right )}{15 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^4,x]

[Out]

(a^4*(150*Csc[c + d*x] + 30*Csc[c + d*x]^2 - 20*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 - 3*Csc[c + d*x]^5 - 60*Log
[Sin[c + d*x]] + 60*Sin[c + d*x] + 30*Sin[c + d*x]^2 + 5*Sin[c + d*x]^3))/(15*d)

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Maple [A]  time = 0.096, size = 235, normalized size = 1.6 \begin{align*}{\frac{24\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,d\sin \left ( dx+c \right ) }}+{\frac{64\,{a}^{4}\sin \left ( dx+c \right ) }{5\,d}}+{\frac{24\,{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{32\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{5\,d}}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-4\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{29\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+2\,{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^6*(a+a*sin(d*x+c))^4,x)

[Out]

24/5/d*a^4/sin(d*x+c)*cos(d*x+c)^6+64/5*a^4*sin(d*x+c)/d+24/5/d*a^4*sin(d*x+c)*cos(d*x+c)^4+32/5/d*a^4*cos(d*x
+c)^2*sin(d*x+c)-2/d*a^4/sin(d*x+c)^2*cos(d*x+c)^6-2/d*a^4*cos(d*x+c)^4-4/d*a^4*cos(d*x+c)^2-4*a^4*ln(sin(d*x+
c))/d-29/15/d*a^4/sin(d*x+c)^3*cos(d*x+c)^6-1/d*a^4*cot(d*x+c)^4+2/d*a^4*cot(d*x+c)^2-1/5/d*a^4/sin(d*x+c)^5*c
os(d*x+c)^6

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Maxima [A]  time = 1.04787, size = 162, normalized size = 1.11 \begin{align*} \frac{5 \, a^{4} \sin \left (d x + c\right )^{3} + 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{4} \sin \left (d x + c\right ) + \frac{150 \, a^{4} \sin \left (d x + c\right )^{4} + 30 \, a^{4} \sin \left (d x + c\right )^{3} - 20 \, a^{4} \sin \left (d x + c\right )^{2} - 15 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{5}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*(5*a^4*sin(d*x + c)^3 + 30*a^4*sin(d*x + c)^2 - 60*a^4*log(sin(d*x + c)) + 60*a^4*sin(d*x + c) + (150*a^4
*sin(d*x + c)^4 + 30*a^4*sin(d*x + c)^3 - 20*a^4*sin(d*x + c)^2 - 15*a^4*sin(d*x + c) - 3*a^4)/sin(d*x + c)^5)
/d

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Fricas [A]  time = 1.6116, size = 482, normalized size = 3.3 \begin{align*} \frac{5 \, a^{4} \cos \left (d x + c\right )^{8} - 80 \, a^{4} \cos \left (d x + c\right )^{6} + 360 \, a^{4} \cos \left (d x + c\right )^{4} - 480 \, a^{4} \cos \left (d x + c\right )^{2} + 192 \, a^{4} - 60 \,{\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 15 \,{\left (2 \, a^{4} \cos \left (d x + c\right )^{6} - 5 \, a^{4} \cos \left (d x + c\right )^{4} + 6 \, a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/15*(5*a^4*cos(d*x + c)^8 - 80*a^4*cos(d*x + c)^6 + 360*a^4*cos(d*x + c)^4 - 480*a^4*cos(d*x + c)^2 + 192*a^4
- 60*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(1/2*sin(d*x + c))*sin(d*x + c) - 15*(2*a^4*cos(d*x
+ c)^6 - 5*a^4*cos(d*x + c)^4 + 6*a^4*cos(d*x + c)^2 - 2*a^4)*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x
+ c)^2 + d)*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**6*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.28392, size = 181, normalized size = 1.24 \begin{align*} \frac{5 \, a^{4} \sin \left (d x + c\right )^{3} + 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{4} \sin \left (d x + c\right ) + \frac{137 \, a^{4} \sin \left (d x + c\right )^{5} + 150 \, a^{4} \sin \left (d x + c\right )^{4} + 30 \, a^{4} \sin \left (d x + c\right )^{3} - 20 \, a^{4} \sin \left (d x + c\right )^{2} - 15 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{5}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/15*(5*a^4*sin(d*x + c)^3 + 30*a^4*sin(d*x + c)^2 - 60*a^4*log(abs(sin(d*x + c))) + 60*a^4*sin(d*x + c) + (13
7*a^4*sin(d*x + c)^5 + 150*a^4*sin(d*x + c)^4 + 30*a^4*sin(d*x + c)^3 - 20*a^4*sin(d*x + c)^2 - 15*a^4*sin(d*x
+ c) - 3*a^4)/sin(d*x + c)^5)/d